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Communications in Mathematical Physics

, Volume 188, Issue 2, pp 267–304 | Cite as

Quantum Integrable Models and Discrete Classical Hirota Equations

  • I. Krichever
  • O. Lipan
  • P. Wiegmann
  • A. Zabrodin

Abstract:

The standard objects of quantum integrable systems are identified with elements of classical nonlinear integrable difference equations. The functional relation for commuting quantum transfer matrices of quantum integrable models is shown to coincide with classical Hirota's bilinear difference equation. This equation is equivalent to the completely discretized classical 2D Toda lattice with open boundaries. Elliptic solutions of Hirota's equation give a complete set of eigenvalues of the quantum transfer matrices. Eigenvalues of Baxter's Q-operator are solutions to the auxiliary linear problems for classical Hirota's equation. The elliptic solutions relevant to the Bethe ansatz are studied. The nested Bethe ansatz equations for A k-1 -type models appear as discrete time equations of motions for zeros of classical τ-functions and Baker-Akhiezer functions. Determinant representations of the general solution to bilinear discrete Hirota's equation are analysed and a new determinant formula for eigenvalues of the quantum transfer matrices is obtained. Difference equations for eigenvalues of the Q-operators which generalize Baxter's three-term TQ-relation are derived.

Keywords

Difference Equation Open Boundary Time Equation Standard Object Discrete Classical 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 1997

Authors and Affiliations

  • I. Krichever
    • 1
  • O. Lipan
    • 2
  • P. Wiegmann
    • 3
  • A. Zabrodin
    • 4
  1. 1.Department of Mathematics, Columbia University, New York, NY 10027, USA and Landau Institute for Theoretical Physics, Kosygina str. 2, 117940 Moscow, RussiaRU
  2. 2.James Franck Institute of the University of Chicago, 5640 S.Ellis Avenue, Chicago, IL 60637, USAUS
  3. 3.James Franck Institute and Enrico Fermi Institute of the University of Chicago, 5640 S.Ellis Avenue, Chicago, IL 60637, USA and Landau Institute for Theoretical PhysicsUS
  4. 4.Joint Institute of Chemical Physics, Kosygina str. 4, 117334, Moscow, Russia and ITEP, 117259, Moscow, RussiaRU

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